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# A Note on Quaternionic Hyperbolic Ideal Triangle Groups

Published:2016-02-16
Printed: Jun 2016
• Wensheng Cao,
School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, P.R. China
• Xiaolin Huang,
School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, P.R. China
 Format: LaTeX MathJax PDF

## Abstract

In this paper, the quaternionic hyperbolic ideal triangle groups are parameterized by a real one-parameter family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{h}}^2$ forming this ideal triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is not a discrete embedding, and if $s \leq \sqrt{35}$ then $\phi_s$ is a discrete embedding.
 Keywords: quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant
 MSC Classifications: 20F67 - Hyperbolic groups and nonpositively curved groups 22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 30F40 - Kleinian groups [See also 20H10]

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